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# Functional Logic • Inquiry and Analogy

Author: Jon Awbrey

This report discusses C.S. Peirce's treatment of analogy, placing it in relation to his overall theory of inquiry.  We begin by introducing three basic types of reasoning Peirce adopted from classical logic.  In Peirce's analysis both inquiry and analogy are complex programs of logical inference which develop through stages of these three types, although normally in different orders.

Note on notation.  The discussion to follow uses logical conjunctions, expressed in the form of concatenated tuples ${\displaystyle e_{1}~\ldots ~e_{k},}$ and minimal negation operations, expressed in the form of bracketed tuples ${\displaystyle {\texttt {(}}e_{1}{\texttt {,}}\ldots {\texttt {,}}e_{k}{\texttt {)}},}$ as the principal expression-forming operations of a calculus for boolean-valued functions, that is, for propositions.  The expressions of this calculus parse into data structures whose underlying graphs are called cacti by graph theorists.  Hence the name cactus language for this dialect of propositional calculus.

## Three Types of Reasoning

### Types of Reasoning in Aristotle

Figure 1 gives a quick overview of traditional terminology we’ll have occasion to refer to as discussion proceeds.

 ${\displaystyle {\text{Figure 1. Types of Reasoning in Aristotle}}}$

### Types of Reasoning in C.S. Peirce

Peirce gives one of his earliest treatments of the three types of reasoning in his Harvard Lectures of 1865 “On the Logic of Science”.  There he shows how the same proposition may be reached from three directions, as the result of an inference in each of the three modes.

 We have then three different kinds of inference: Deduction or inference à priori, Induction or inference à particularis, Hypothesis or inference à posteriori. (Peirce, CE 1, p. 267).
 If I reason that certain conduct is wise because it has a character which belongs only to wise things, I reason à priori. If I think it is wise because it once turned out to be wise, that is, if I infer that it is wise on this occasion because it was wise on that occasion, I reason inductively [à particularis]. But if I think it is wise because a wise man does it, I then make the pure hypothesis that he does it because he is wise, and I reason à posteriori. (Peirce, CE 1, p. 180).

Suppose we make the following assignments.

${\displaystyle {\begin{array}{lll}\mathrm {A} &=&{\text{Wisdom}}\\\mathrm {B} &=&{\text{a certain character}}\\\mathrm {C} &=&{\text{a certain conduct}}\\\mathrm {D} &=&{\text{done by a wise man}}\\\mathrm {E} &=&{\text{a certain occasion}}\end{array}}}$

Recognizing a little more concreteness will aid understanding, let us make the following substitutions in Peirce's example.

${\displaystyle {\begin{array}{lllll}\mathrm {B} &=&{\text{Benevolence}}&=&{\text{a certain character}}\\\mathrm {C} &=&{\text{Contributes to Charity}}&=&{\text{a certain conduct}}\\\mathrm {E} &=&{\text{Earlier today}}&=&{\text{a certain occasion}}\end{array}}}$

The converging operation of all three reasonings is shown in Figure 2.

 ${\displaystyle {\text{Figure 2. A Triply Wise Act}}}$

The common proposition concluding each argument is AC, contributing to charity is wise.

• Deduction could have obtained the Fact AC from the Rule AB, benevolence is wisdom, along with the Case BC, contributing to charity is benevolent.
• Induction could have gathered the Rule AC, contributing to charity is exemplary of wisdom, from the Fact AE, the act of earlier today is wise, along with the Case CE, the act of earlier today was an instance of contributing to charity.
• Abduction could have guessed the Case AC, contributing to charity is explained by wisdom, from the Fact DC, contributing to charity is done by this wise man, and the Rule DA, everything wise is done by this wise man.
Thus, a wise man, who does all the wise things there are to do, may nonetheless contribute to charity for no good reason and even be charitable to a fault.  But on seeing the wise man contribute to charity it is natural to think charity may well be the mark of his wisdom, in essence, that wisdom is the reason he contributes to charity.

### Comparison of the Analyses

The next two Figures will be of use when we turn to comparing the three types of inference as they appear in the respective analyses of Aristotle and Peirce.

 ${\displaystyle {\text{Figure 3. Types of Reasoning in Transition}}}$

 ${\displaystyle {\text{Figure 4. Types of Reasoning in Peirce}}}$

### Aristotle's “Apagogy” • Abductive Reasoning as Problem Reduction

Peirce's notion of abductive reasoning was derived from Aristotle's treatment of it in the Prior Analytics. Aristotle's discussion begins with an example that may appear incidental, but the question and its analysis are echoes of an important investigation pursued in one of Plato's Dialogues, the Meno.  This inquiry is concerned with the possibility of knowledge and the relationship between knowledge and virtue, or between their objects, the true and the good.  It is not just because it forms a recurring question in philosophy, but because it preserves a certain correspondence between its form and its content, that we shall find this example increasingly relevant to our study.

A couple of notes on the reading may be helpful. The Greek text seems to imply a geometric diagram, in which directed line segments AB, BC, AC are used to indicate logical relations between pairs of the terms in A, B, C. We have two options for reading these line labels, either as implications or as subsumptions, as in the following two paradigms for interpretation.

Here, “X subsumes Y” means “X applies to all Y”, or “X is predicated of all Y”.  When there is no danger of confusion we may write this as “XY”.

 We have Reduction (απαγωγη, abduction): (1) when it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet nevertheless is more probable or not less probable than the conclusion; or (2) if there are not many intermediate terms between the last and the middle; for in all such cases the effect is to bring us nearer to knowledge.       (1) E.g., let A stand for “that which can be taught”, B for “knowledge”, and C for “morality”. Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if BC is not less probable or is more probable than AC, we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true.       (2) Or again we have reduction if there are not many intermediate terms between B and C; for in this case too we are brought nearer to knowledge. E.g., suppose that D is “to square”, E “rectilinear figure”, and F “circle”. Assuming that between E and F there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of lunules — we should approximate to knowledge. (Aristotle, “Prior Analytics” 2.25)

The method of abductive reasoning bears a close relation to the sense of reduction in which we speak of one question reducing to another.  The question being asked is “Can virtue be taught?”  The type of answer which develops is the following.

If virtue is a form of understanding, and if we are willing to grant that understanding can be taught, then virtue can be taught.  In this way of approaching the problem, by detour and indirection, the form of abductive reasoning is used to shift the attack from the original question, whether virtue can be taught, to the hopefully easier question, whether virtue is a form of understanding.

The logical structure of the process of hypothesis formation in the first example follows the pattern of “abduction to a case”, whose abstract form is shown in Figure 5.

 ${\displaystyle {\text{Figure 5. Teachability, Understanding, Virtue}}}$

The sense of the Figure is explained by the following assignments.

Abduction from a Fact to a Case proceeds according to the following schema.

${\displaystyle {\begin{array}{l}~{\text{Fact:}}~V\Rightarrow T?\\~{\text{Rule:}}~U\Rightarrow T.\\{\overline {~~~~~~~~~~~~~~~~~~~~~~}}\\~{\text{Case:}}~V\Rightarrow U?\end{array}}}$

### Aristotle's “Paradigm” • Reasoning by Analogy or Example

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word παραδειγμα, from which comes the English word paradigm.  In its original sense the word suggests a kind of “side‑show”, or a parallel comparison of cases.

 We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.       E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad. (Aristotle, “Prior Analytics” 2.24)
 ${\displaystyle {\text{Figure 6. Aristotle's “Paradigm”}}}$

### Peirce's Formulation of Analogy

Next we look at a couple of ways Peirce analyzed analogical inference.  Version 1 —

Note.  A few changes in Peirce's notation have been made to facilitate comparison between the two versions.

#### Version 1. “On the Natural Classification of Arguments” (1867)

 The formula of analogy is as follows:   ${\displaystyle S^{\prime },S^{\prime \prime },~{\text{and}}~S^{\prime \prime \prime }}$ are taken at random from such a class that their characters at random are such as ${\displaystyle {P^{\prime },P^{\prime \prime },P^{\prime \prime \prime }}.}$ ${\displaystyle {\begin{matrix}T~{\text{is}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime }~{\text{are}}~Q;\\[4pt]\therefore T~{\text{is}}~Q.\end{matrix}}}$ Such an argument is double. It combines the two following: ${\displaystyle {\begin{matrix}1.\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime }~{\text{are taken as being}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime }~{\text{are}}~Q;\\[4pt]\therefore ~({\text{By induction}})~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime }~{\text{is}}~Q,\\[4pt]T~{\text{is}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime };\\[4pt]\therefore ~({\text{Deductively}})~T~{\text{is}}~Q.\end{matrix}}}$ ${\displaystyle {\begin{matrix}2.\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime }~{\text{are, for instance,}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },\\[4pt]T~{\text{is}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime };\\[4pt]\therefore ~({\text{By hypothesis}})~T~{\text{has the common characters of}}~S^{\prime },S^{\prime \prime },S^{\prime \prime \prime },\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime }~{\text{are}}~Q;\\[4pt]\therefore ~({\text{Deductively}})~T~{\text{is}}~Q.\end{matrix}}}$ Owing to its double character, analogy is very strong with only a moderate number of instances. (Peirce, CP 2.513; CE 2, 46–47)

Figure 7 shows the logical relationships involved in the above analysis.

 ${\displaystyle {\text{Figure 7. Peirce's Formulation of Analogy (Version 1)}}}$

#### Version 2. “A Theory of Probable Inference” (1883)

Peirce gave a more complex formulation of analogy at a later date.  Version 2 —

 The formula of the analogical inference presents, therefore, three premisses, thus:   ${\displaystyle S^{\prime },S^{\prime \prime },S^{\prime \prime \prime },}$ are a random sample of some undefined class, ${\displaystyle X,}$ of whose characters ${\displaystyle P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },}$ are samples, ${\displaystyle {\begin{matrix}T~{\text{is}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime };\\[4pt]S^{\prime },S^{\prime \prime },S^{\prime \prime \prime },~{\text{are}}~Q\operatorname {'s} ;\\[4pt]{\text{Hence,}}~T~{\text{is a}}~Q.\end{matrix}}}$ We have evidently here an induction and an hypothesis followed by a deduction; thus: ${\displaystyle {\begin{array}{l|l}{\text{Every}}~X~{\text{is, for example,}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },~{\text{etc.}},&S^{\prime },S^{\prime \prime },S^{\prime \prime \prime },~{\text{etc., are samples of the}}~X\operatorname {'s} ,\\[4pt]T~{\text{is found to be}}~P^{\prime },P^{\prime \prime },P^{\prime \prime \prime },~{\text{etc.}};&S^{\prime },S^{\prime \prime },S^{\prime \prime \prime },~{\text{etc., are found to be}}~Q\operatorname {'s} ;\\[4pt]{\text{Hence, hypothetically,}}~T~{\text{is an}}~X.&{\text{Hence, inductively, every}}~X~{\text{is a}}~Q.\end{array}}}$ ${\displaystyle {\text{Hence, deductively,}}~T~{\text{is a}}~Q.}$ (Peirce, CP 2.733)

Figure 8 shows the logical relationships involved in the above analysis.

 ${\displaystyle {\text{Figure 8. Peirce's Formulation of Analogy (Version 2)}}}$

### Dewey's “Sign of Rain” • An Example of Inquiry

To illustrate the place of the sign relation in inquiry we begin with Dewey's elegant and simple example of reflective thinking in everyday life.

 A man is walking on a warm day.  The sky was clear the last time he observed it;  but presently he notes, while occupied primarily with other things, that the air is cooler.  It occurs to him that it is probably going to rain;  looking up, he sees a dark cloud between him and the sun, and he then quickens his steps.  What, if anything, in such a situation can be called thought?  Neither the act of walking nor the noting of the cold is a thought.  Walking is one direction of activity;  looking and noting are other modes of activity.  The likelihood that it will rain is, however, something suggested.  The pedestrian feels the cold;  he thinks of clouds and a coming shower. (Dewey 1991, 6–7)

#### Inquiry and Interpretation

In Dewey's narrative we can identify the characters of the sign relation as follows.  Coolness is a Sign of the Object rain, and the Interpretant is the thought of the rain's likelihood.  In his 1910 description of reflective thinking Dewey distinguishes two phases, “a state of perplexity, hesitation, doubt” and “an act of search or investigation” (Dewey 1991, 9), comprehensive stages which are further refined in his later model of inquiry.  In this example, reflection is the act of the interpreter which establishes a fund of connections between the sensory shock of coolness and the objective danger of rain, by way of his impression that rain is likely.  But reflection is more than irresponsible speculation.  In reflection the interpreter acts to charge or defuse the thought of rain (the probability of rain in thought) by seeking other signs which this thought implies and evaluating the thought according to the results of this search.

Figure 9 illustrates Dewey's “Sign of Rain” example, tracing the structure and function of the sign relation as it informs the activity of inquiry, including both the movements of surprise explanation and intentional action.  The dyadic faces of the sign relation are labeled with a few of the loosest terms that apply, indicating the “significance” of signs for eventual occurrences and the “correspondence&rdqu; of ideas with external orientations.  Nothing essential is meant by these dyadic role distinctions, since it is only in special or degenerate cases that their shadowy projections can maintain enough information to determine the original sign relation.

 ${\displaystyle {\text{Figure 9. Dewey's “Sign of Rain” Example}}}$

#### Inquiry and Inference

If we follow Dewey's story far enough to consider the import of thought for action, we realize that the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, seeking shelter in time to escape the rain — all of these acts form a series of further interpretants, contingent on the active causes of the individual, for the originally recognized signs of rain and for the first impressions of the actual case.  Just as critical reflection develops the associated and alternative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person's belief in it.

Figure 10 charts the progress of inquiry in this example according to the three stages of reasoning identified by Peirce.

 ${\displaystyle {\text{Figure 10. The Cycle of Inquiry}}}$
1. Abduction. The first, faltering step into the cycle of inquiry is taken through the flexion of abductive reasoning.  The fact CA, the coolness of the air in the pedestrian's current situation, brings into play from his worldly experience (or from other kinds of background knowledge) the rule BA, that a chill in the air is a feature of situations that betoken rain.  This fact and this rule, working in tandem, precipitate a plausible explanation for the observed phenomena.  The hiker abduces the case CB, that bodes for rain in the current situation.
2. Deduction. …
3. Induction. …

In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps:

1.  The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule.

 Fact: C ⇒ A, In the Current situation the Air is cool. Rule: B ⇒ A, Just Before it rains, the Air is cool. Case: C ⇒ B, The Current situation is just Before it rains.

2.  The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact.

 Case: C ⇒ B, The Current situation is just Before it rains. Rule: B ⇒ D, Just Before it rains, a Dark cloud will appear. Fact: C ⇒ D, In the Current situation, a Dark cloud will appear.

This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start.

## Functional Conception of Quantification Theory

Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements.  The mere act of writing quantified formulas like ${\displaystyle \forall _{x\in X}f(x)}$ and ${\displaystyle \exists _{x\in X}f(x)}$ involves a subscription to such notions, as shown by the membership relations invoked in their indices.  As we reflect more critically on the conventional assumptions in the light of pragmatic and constructive principles, however, they begin to appear as problematic hypotheses whose warrants are not beyond question, as projects of exhaustive determination overreaching the powers of finite information and control to manage.  Thus it is worth considering how the scene of quantification theory might be shifted nearer to familiar ground, toward the predicates themselves which represent our continuing acquaintance with phenomena.

### Higher Order Propositional Expressions

If functions of type ${\displaystyle X\to \mathbb {B} }$ are propositions about things in ${\displaystyle X}$ then functions of type ${\displaystyle (X\to \mathbb {B} )\to \mathbb {B} }$ are propositions about propositions about things in ${\displaystyle X,}$ the first in a series of higher order propositions based on ${\displaystyle X.}$

To ground this inquiry in concrete material, let us begin with a consideration of higher order propositional expressions, in particular, those stemming from propositions on 1 and 2 variables.

#### Higher Order Propositions and Logical Operators (n = 1)

A higher order proposition is a proposition about propositions.  If the original order of propositions is a class of indicator functions ${\displaystyle f:X\to \mathbb {B} }$ then the next higher order of propositions consists of maps of type ${\displaystyle m:(X\to \mathbb {B} )\to \mathbb {B} .}$

For example, consider the case where ${\displaystyle X=\mathbb {B} .}$  There are exactly four propositions one can make about the elements of ${\displaystyle X.}$  Each proposition has the concrete type ${\displaystyle f:X\to \mathbb {B} }$ and the abstract type ${\displaystyle f:\mathbb {B} \to \mathbb {B} .}$  From that beginning there are exactly sixteen higher order propositions one can make about the initial set of four propositions.  Each higher order proposition has the abstract type ${\displaystyle m:(\mathbb {B} \to \mathbb {B} )\to \mathbb {B} .}$

Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion.

• Columns 1 and 2 taken together present a form of truth table for the four propositions ${\displaystyle f:\mathbb {B} \to \mathbb {B} .}$  Column 1 displays the names of the propositions ${\displaystyle f_{i},}$ for ${\displaystyle i}$ = 1 to 4, while the entries in Column 2 show the value each proposition takes on the argument value listed in the corresponding column head.
• Column 3 displays one of the more usual expressions for the proposition in question.
• The last sixteen columns are headed by a series of conventional names for the higher order propositions, also known as the measures ${\displaystyle m_{j},}$ for ${\displaystyle j}$ = 0 to 15.  The entries in the body of the Table show the value each measure assigns to each proposition ${\displaystyle f_{i}.}$

 ${\displaystyle {\text{Table 11. Higher Order Propositions}}~(n=1)}$

Table 12 presents a series of interpretive categories for the higher order propositions in Table 11.  I'll postpone further discussion of those until we get beyond the 1-dimensional case.  The lower dimensional cases tend to be condensed or degenerate in their structures and their full significance becomes almost automatically easier to see as soon we get two variables into the mix.

 ${\displaystyle {\text{Table 12. Interpretive Categories for Higher Order Propositions}}~(n=1)}$

#### Higher Order Propositions and Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let's first consider the universe of discourse ${\displaystyle X^{\bullet }=[{\mathcal {X}}]=[x_{1},x_{2}]=[u,v],}$ based on two logical features or boolean variables ${\displaystyle u}$ and ${\displaystyle v.}$

The universe of discourse ${\displaystyle {X^{\bullet }}}$ consists of two parts, a set of points and a set of propositions.

The points of ${\displaystyle {X^{\bullet }}}$ form the space:

${\displaystyle {\begin{matrix}X&=&\langle {\mathcal {X}}\rangle &=&\langle u,v\rangle &=&\{(u,v)\}&\cong &\mathbb {B} ^{2}.\end{matrix}}}$

Each point in ${\displaystyle {X}}$ may be indicated by means of a singular proposition, that is, a proposition which describes it uniquely.  This form of representation leads to the following enumeration of points.

${\displaystyle {\begin{matrix}X&=&\{~{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}~,~{\texttt {(}}u{\texttt {)}}~v~,~u~{\texttt {(}}v{\texttt {)}}~,~u~v~\}&\cong &\mathbb {B} ^{2}.\end{matrix}}}$

Each point in ${\displaystyle X}$ may also be described by means of its coordinates, that is, by the ordered pair of values in ${\displaystyle \mathbb {B} }$ which the coordinate propositions ${\displaystyle u}$ and ${\displaystyle v}$ take on that point.  This form of representation leads to the following enumeration of points.

${\displaystyle {\begin{matrix}X&=&\{\ (0,0),\ (0,1),\ (1,0),\ (1,1)\ \}&\cong &\mathbb {B} ^{2}.\end{matrix}}}$

The propositions of ${\displaystyle {X^{\bullet }}}$ form the space:

${\displaystyle {\begin{matrix}X^{\uparrow }&=&(X\to \mathbb {B} )&=&\{f:X\to \mathbb {B} \}&\cong &(\mathbb {B} ^{2}\to \mathbb {B} ).\end{matrix}}}$

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.

The next higher order universe of discourse built on ${\displaystyle {X^{\bullet }}}$ is ${\displaystyle X^{\bullet 2}=[X^{\bullet }]=[[u,v]],}$ which may be developed in the following way.  The propositions of ${\displaystyle {X^{\bullet }}}$ become the points of ${\displaystyle X^{\bullet 2},}$ and the mappings of the type ${\displaystyle m:(X\to \mathbb {B} )\to \mathbb {B} }$ become the propositions of ${\displaystyle X^{\bullet 2}.}$  In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form ${\displaystyle w:(\mathbb {B} ^{2}\to \mathbb {B} )^{k}\to \mathbb {B} .}$

To save a few words in the remainder of this discussion, I will use the terms measure and qualifier to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of ${\displaystyle [u,v]}$ may be regarded as a gallery of sixteen venn diagrams, while the measures ${\displaystyle m:(X\to \mathbb {B} )\to \mathbb {B} }$ are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge ${\displaystyle m_{j}}$ partitions the gallery of pictures into two aesthetic portions, the pictures ${\displaystyle m_{j}^{-1}(1)}$ that ${\displaystyle m_{j}}$ likes and the pictures ${\displaystyle m_{j}^{-1}(0)}$ that ${\displaystyle m_{j}}$ dislikes.

There are ${\displaystyle 2^{16}=65536}$ measures of the form ${\displaystyle m:(\mathbb {B} ^{2}\to \mathbb {B} )\to \mathbb {B} .}$  Table 13 shows the first 24 of their number in the style of higher order truth table used above.  The column headed ${\displaystyle m_{j}}$ shows the value of the measure ${\displaystyle m_{j}}$ on each of the propositions ${\displaystyle f_{i}:\mathbb {B} ^{2}\to \mathbb {B} }$ for ${\displaystyle i}$ = 0 to 15.  The arrangement of measures in the order indicated will be called their standard ordering.  In this scheme of things, the index ${\displaystyle j}$ of the measure ${\displaystyle m_{j}}$ is the decimal equivalent of the bit string in the corresponding column of the Table, reading the binary digits in order from bottom to top.

 ${\displaystyle {\text{Table 13. Higher Order Propositions}}~(n=2)}$

### Umpire Operators

The ${\displaystyle 2^{16}}$ measures of type ${\displaystyle (\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} }$ present a formidable array of propositions about propositions about 2-dimensional universes of discourse.  The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

${\displaystyle {\begin{matrix}\Upsilon :(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )^{2}\to \mathbb {B} &&{\text{and}}&&\Upsilon _{1}:(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} ,\end{matrix}}}$

referred to as the relative and absolute umpire operators, respectively.  If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

Let ${\displaystyle X=\langle u,v\rangle }$ be a two-dimensional boolean space, ${\displaystyle X\cong \mathbb {B} \times \mathbb {B} ,}$ generated by two boolean variables or logical features ${\displaystyle u}$ and ${\displaystyle v.}$

Given an ordered pair of propositions ${\displaystyle e,f:\langle u,v\rangle \to \mathbb {B} }$ as arguments, the relative umpire operator reports the value ${\displaystyle 1}$ if the first implies the second, otherwise it reports the value ${\displaystyle 0.}$

${\displaystyle {\begin{matrix}\Upsilon (e,f)=1&&{\text{if and only if}}&&e\Rightarrow f\end{matrix}}}$

Expressing it another way:

${\displaystyle {\begin{matrix}\Upsilon (e,f)=1&&\iff &&{\texttt {(}}e{\texttt {(}}f{\texttt {))}}=1\end{matrix}}}$

In writing this, however, it is important to observe that the ${\displaystyle 1}$ appearing on the left side and the ${\displaystyle 1}$ appearing on the right side of the logical equivalence have different meanings.  Filling in the details, we have the following.

${\displaystyle {\begin{matrix}\Upsilon (e,f)=1\in \mathbb {B} &&\iff &&{\texttt {(}}e{\texttt {(}}f{\texttt {))}}=1:\langle u,v\rangle \to \mathbb {B} \end{matrix}}}$

Writing types as subscripts and using the fact that ${\displaystyle X=\langle u,v\rangle ,}$ it is possible to express this a little more succinctly as follows.

${\displaystyle {\begin{matrix}\Upsilon (e,f)=1_{\mathbb {B} }&&\iff &&{\texttt {(}}e{\texttt {(}}f{\texttt {))}}=1_{X\to \mathbb {B} }\end{matrix}}}$

Finally, it is often convenient to write the first argument as a subscript.  Thus we have the following equation.

${\displaystyle {\begin{matrix}\Upsilon _{e}(f)&=&\Upsilon (e,f).\end{matrix}}}$

The absolute umpire operator, also known as the umpire measure, is a higher order proposition ${\displaystyle \Upsilon _{1}:(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} }$ defined by the equation ${\displaystyle \Upsilon _{1}(f)=\Upsilon (1,f).}$  In this case the subscript ${\displaystyle 1}$ on the left and the argument ${\displaystyle 1}$ on the right both refer to the constant proposition ${\displaystyle 1:\mathbb {B} \times \mathbb {B} \to \mathbb {B} .}$  In most settings where ${\displaystyle \Upsilon _{1}}$ is applied to arguments it is safe to omit the subscript ${\displaystyle 1}$ since the number of arguments indicates which type of operator is meant.  Thus, we have the following identities and equivalents.

${\displaystyle {\begin{matrix}\Upsilon f=\Upsilon _{1}(f)=1_{\mathbb {B} }&\iff &{\texttt {(}}1{\texttt {(}}f{\texttt {))}}=\mathbf {1} &\iff &f=1_{\mathbb {B} \times \mathbb {B} \to \mathbb {B} }\end{matrix}}}$

The umpire measure ${\displaystyle \Upsilon _{1}}$ is defined at the level of boolean functions as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level.  In that interpretation ${\displaystyle \Upsilon _{1}}$ recognizes theorems of the propositional calculus over ${\displaystyle [u,v],}$ giving a score of ${\displaystyle 1}$ to tautologies and a score of ${\displaystyle 0}$ to everything else, regarding all contingent statements as no better than falsehoods.

One remark in passing for those who might prefer an alternative definition.  If we had originally taken ${\displaystyle \Upsilon }$ to mean the absolute measure then the relative measure could have been defined as ${\displaystyle \Upsilon _{e}f=\Upsilon {\texttt {(}}e{\texttt {(}}f{\texttt {))}}.}$

## Measure for Measure

Let us define two families of measures,

${\displaystyle \alpha _{i},\beta _{i}:(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} ~{\text{for}}~i=0~{\text{to}}~15,}$

by means of the following equations:

${\displaystyle {\begin{matrix}\alpha _{i}f&=&\Upsilon (f_{i},f)&=&\Upsilon (f_{i}\Rightarrow f),\\[6pt]\beta _{i}f&=&\Upsilon (f,f_{i})&=&\Upsilon (f\Rightarrow f_{i}).\end{matrix}}}$

Table 14 shows the value of each ${\displaystyle \alpha _{i}}$ on each of the 16 boolean functions ${\displaystyle f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} .}$  In terms of the implication ordering on the 16 functions, ${\displaystyle \alpha _{i}f=1}$ says that ${\displaystyle f}$ is above or identical to ${\displaystyle f_{i}}$ in the implication lattice, that is, ${\displaystyle f\geq f_{i}}$ in the implication ordering.

 ${\displaystyle {\text{Table 14. Qualifiers of the Implication Ordering}}~\alpha _{i}f=\Upsilon (f_{i},f)}$

Table 15 shows the value of each ${\displaystyle \beta _{i}}$ on each of the 16 boolean functions ${\displaystyle f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} .}$  In terms of the implication ordering on the 16 functions, ${\displaystyle \beta _{i}f=1}$ says that ${\displaystyle f}$ is below or identical to ${\displaystyle f_{i}}$ in the implication lattice, that is, ${\displaystyle f\leq f_{i}}$ in the implication ordering.

 ${\displaystyle {\text{Table 15. Qualifiers of the Implication Ordering}}~\beta _{i}f=\Upsilon (f,f_{i})}$

Applied to a given proposition ${\displaystyle f,}$ the qualifiers ${\displaystyle \alpha _{i}}$ and ${\displaystyle \beta _{i}}$ tell whether ${\displaystyle f}$ is above ${\displaystyle f_{i}}$ or below ${\displaystyle f_{i},}$ respectively, in the implication ordering.  By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.

${\displaystyle {\begin{array}{*{8}{r}}\alpha _{0}f=1&\mathrm {iff} &f_{0}\Rightarrow f&\mathrm {iff} &0\Rightarrow f,&\mathrm {hence} &\alpha _{0}f=1&\mathrm {for~all} ~f.\\[4pt]\alpha _{15}f=1&\mathrm {iff} &f_{15}\Rightarrow f&\mathrm {iff} &1\Rightarrow f,&\mathrm {hence} &\alpha _{15}f=1&\mathrm {iff} ~f=1.\\[4pt]\beta _{0}f=1&\mathrm {iff} &f\Rightarrow f_{0}&\mathrm {iff} &f\Rightarrow 0,&\mathrm {hence} &\beta _{0}f=1&\mathrm {iff} ~f=0.\\[4pt]\beta _{15}f=1&\mathrm {iff} &f\Rightarrow f_{15}&\mathrm {iff} &f\Rightarrow 1,&\mathrm {hence} &\beta _{15}f=1&\mathrm {for~all} ~f.\end{array}}}$

Expressed in terms of the propositional forms they value positively, ${\displaystyle \alpha _{0}=\beta _{15}}$ is a totally indiscriminate measure, accepting all propositions ${\displaystyle f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} ,}$ whereas ${\displaystyle \alpha _{15}}$ and ${\displaystyle \beta _{0}}$ are measures valuing the constant propositions ${\displaystyle 1:\mathbb {B} \times \mathbb {B} \to \mathbb {B} }$ and ${\displaystyle 0:\mathbb {B} \times \mathbb {B} \to \mathbb {B} ,}$ respectively, above all others.

Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions satisfying the umpires in question.

${\displaystyle {\begin{matrix}[|\alpha _{i}|]&=&\alpha _{i}^{-1}(1),\\[6pt][|\beta _{i}|]&=&\beta _{i}^{-1}(1),\\[6pt][|\Upsilon _{p}|]&=&\Upsilon _{p}^{-1}(1).\end{matrix}}}$

### Extending the Existential Interpretation to Quantificational Logic

One of the resources we have for this work is a formal calculus based on C.S. Peirce's logical graphs.  For now we'll adopt the existential interpretation of that calculus, fixing the meanings of logical constants and connectives at the core level of propositional logic.  To build on that core we'll need to extend the existential interpretation to encompass the analysis of quantified propositions, or quantifications.  That in turn will take developing two further capacities of our calculus.  On the formal side we'll need to consider higher order functional types, continuing our earlier venture above.  In terms of content we'll need to consider new species of elemental or singular propositions.

Let us return to the 2-dimensional universe ${\displaystyle X^{\bullet }=[u,v].}$  A bridge between propositions and quantifications is afforded by a set of measures or qualifiers ${\displaystyle \ell _{ij}:(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} }$ defined by the following equations.

${\displaystyle {\begin{array}{*{11}{l}}\ell _{00}f&=&\ell _{{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}}f&=&\alpha _{1}f&=&\Upsilon _{{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}}f&=&\Upsilon _{{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\,\Rightarrow \,f}&=&f~{\text{likes}}~{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\\ell _{01}f&=&\ell _{{\texttt {(}}u{\texttt {)}}v}f&=&\alpha _{2}f&=&\Upsilon _{{\texttt {(}}u{\texttt {)}}v}f&=&\Upsilon _{{\texttt {(}}u{\texttt {)}}v\,\Rightarrow \,f}&=&f~{\text{likes}}~{\texttt {(}}u{\texttt {)}}v\\\ell _{10}f&=&\ell _{u{\texttt {(}}v{\texttt {)}}}f&=&\alpha _{4}f&=&\Upsilon _{u{\texttt {(}}v{\texttt {)}}}f&=&\Upsilon _{u{\texttt {(}}v{\texttt {)}}\,\Rightarrow \,f}&=&f~{\text{likes}}~u{\texttt {(}}v{\texttt {)}}\\\ell _{11}f&=&\ell _{u\,v}f&=&\alpha _{8}f&=&\Upsilon _{u\,v}f&=&\Upsilon _{u\,v\,\Rightarrow \,f}&=&f~{\text{likes}}~u\,v\end{array}}}$

A higher order proposition ${\displaystyle \ell _{ij}:(\mathbb {B} \times \mathbb {B} \to \mathbb {B} )\to \mathbb {B} }$ tells us something about the proposition ${\displaystyle f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} ,}$ namely, which elements in the space of type ${\displaystyle \mathbb {B} \times \mathbb {B} }$ are assigned a positive value by ${\displaystyle f.}$  Taken together, the ${\displaystyle \ell _{ij}}$ operators give us a way to express many useful observations about the propositions in ${\displaystyle X^{\bullet }=[u,v].}$  Figure 16 summarizes the action of the ${\displaystyle \ell _{ij}}$ operators on the propositions of type ${\displaystyle f:\mathbb {B} \times \mathbb {B} \to \mathbb {B} .}$

 ${\displaystyle {\text{Figure 16. Higher Order Universe of Discourse}}~\left[\ell _{00},\ell _{01},\ell _{10},\ell _{11}\right]\subseteq \left[\left[u,v\right]\right]}$

### Application of Higher Order Propositions to Quantification Theory

Our excursion into the expanding landscape of higher order propositions has come round to the point where we can begin to open up new perspectives on quantificational logic.

Though it may be all the same from a purely formal point of view, it does serve intuition to adopt a slightly different interpretation for the two-valued space we take as the target of our basic indicator functions.  In that spirit we declare a novel type of existence-valued functions ${\displaystyle f:\mathbb {B} ^{k}\to \mathbb {E} }$ where ${\displaystyle \mathbb {E} =\{-e,+e\}=\{\mathrm {empty} ,\mathrm {existent} \}}$ is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse.  As usual, we won't be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough.  With that interpretation in mind the next few Tables illustrate the correspondence between classical quantification theory and higher order indicator functions.

Table 17 exhibits a fourfold schema of quantified propositional forms traditionally known as a “Square of Opposition”, relating it to a quartet of higher order propositions which, depending on context, are also known as measures, qualifiers, or higher order indicator functions.

 ${\displaystyle {\begin{array}{clcl}\mathrm {A} &{\text{Universal Affirmative}}&{\text{All}}~u~{\text{is}}~v&{\text{Indicator of}}~u{\texttt {(}}v{\texttt {)}}=0\\[4pt]\mathrm {E} &{\text{Universal Negative}}&{\text{All}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}&{\text{Indicator of}}~u\cdot v=0\\[4pt]\mathrm {I} &{\text{Particular Affirmative}}&{\text{Some}}~u~{\text{is}}~v&{\text{Indicator of}}~u\cdot v=1\\[4pt]\mathrm {O} &{\text{Particular Negative}}&{\text{Some}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}&{\text{Indicator of}}~u{\texttt {(}}v{\texttt {)}}=1\end{array}}}$

Table 18 develops the above ideas in further detail, expressing a larger set of quantified propositional forms by means of propositions about propositions.

 ${\displaystyle {\begin{matrix}u\!:\\v\!:\end{matrix}}}$ ${\displaystyle {\begin{matrix}1100\\1010\end{matrix}}}$ ${\displaystyle f}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{11}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{10}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{01}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{00}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{00}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{01}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{10}\\{\text{Some}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{11}\\{\text{Some}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0000}$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ 1 1 1 1 0 0 0 0 ${\displaystyle f_{1}}$ ${\displaystyle 0001}$ ${\displaystyle {\texttt {(}}u{\texttt {)(}}v{\texttt {)}}}$ 1 1 1 0 1 0 0 0 ${\displaystyle f_{2}}$ ${\displaystyle 0010}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}~v}$ 1 1 0 1 0 1 0 0 ${\displaystyle f_{3}}$ ${\displaystyle 0011}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}}$ 1 1 0 0 1 1 0 0 ${\displaystyle f_{4}}$ ${\displaystyle 0100}$ ${\displaystyle u~{\texttt {(}}v{\texttt {)}}}$ 1 0 1 1 0 0 1 0 ${\displaystyle f_{5}}$ ${\displaystyle 0101}$ ${\displaystyle {\texttt {(}}v{\texttt {)}}}$ 1 0 1 0 1 0 1 0 ${\displaystyle f_{6}}$ ${\displaystyle 0110}$ ${\displaystyle {\texttt {(}}u{\texttt {,}}v{\texttt {)}}}$ 1 0 0 1 0 1 1 0 ${\displaystyle f_{7}}$ ${\displaystyle 0111}$ ${\displaystyle {\texttt {(}}u~v{\texttt {)}}}$ 1 0 0 0 1 1 1 0 ${\displaystyle f_{8}}$ ${\displaystyle 1000}$ ${\displaystyle u~v}$ 0 1 1 1 0 0 0 1 ${\displaystyle f_{9}}$ ${\displaystyle 1001}$ ${\displaystyle {\texttt {((}}u{\texttt {,}}v{\texttt {))}}}$ 0 1 1 0 1 0 0 1 ${\displaystyle f_{10}}$ ${\displaystyle 1010}$ ${\displaystyle v}$ 0 1 0 1 0 1 0 1 ${\displaystyle f_{11}}$ ${\displaystyle 1011}$ ${\displaystyle {\texttt {(}}u~{\texttt {(}}v{\texttt {))}}}$ 0 1 0 0 1 1 0 1 ${\displaystyle f_{12}}$ ${\displaystyle 1100}$ ${\displaystyle u}$ 0 0 1 1 0 0 1 1 ${\displaystyle f_{13}}$ ${\displaystyle 1101}$ ${\displaystyle {\texttt {((}}u{\texttt {)}}~v{\texttt {)}}}$ 0 0 1 0 1 0 1 1 ${\displaystyle f_{14}}$ ${\displaystyle 1110}$ ${\displaystyle {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}}$ 0 0 0 1 0 1 1 1 ${\displaystyle f_{15}}$ ${\displaystyle 1111}$ ${\displaystyle {\texttt {((}}~{\texttt {))}}}$ 0 0 0 0 1 1 1 1

Tables 19 and 20 present the same information as Table 18, sorting the rows in different orders to reveal other symmetries in the arrays.

 ${\displaystyle {\begin{matrix}u\!:\\v\!:\end{matrix}}}$ ${\displaystyle {\begin{matrix}1100\\1010\end{matrix}}}$ ${\displaystyle f}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{11}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{10}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{01}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{00}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{00}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{01}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{10}\\{\text{Some}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{11}\\{\text{Some}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0000}$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ 1 1 1 1 0 0 0 0 ${\displaystyle f_{1}}$ ${\displaystyle 0001}$ ${\displaystyle {\texttt {(}}u{\texttt {)(}}v{\texttt {)}}}$ 1 1 1 0 1 0 0 0 ${\displaystyle f_{2}}$ ${\displaystyle 0010}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}~v}$ 1 1 0 1 0 1 0 0 ${\displaystyle f_{4}}$ ${\displaystyle 0100}$ ${\displaystyle u~{\texttt {(}}v{\texttt {)}}}$ 1 0 1 1 0 0 1 0 ${\displaystyle f_{8}}$ ${\displaystyle 1000}$ ${\displaystyle u~v}$ 0 1 1 1 0 0 0 1 ${\displaystyle f_{3}}$ ${\displaystyle 0011}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}}$ 1 1 0 0 1 1 0 0 ${\displaystyle f_{12}}$ ${\displaystyle 1100}$ ${\displaystyle u}$ 0 0 1 1 0 0 1 1 ${\displaystyle f_{6}}$ ${\displaystyle 0110}$ ${\displaystyle {\texttt {(}}u{\texttt {,}}v{\texttt {)}}}$ 1 0 0 1 0 1 1 0 ${\displaystyle f_{9}}$ ${\displaystyle 1001}$ ${\displaystyle {\texttt {((}}u{\texttt {,}}v{\texttt {))}}}$ 0 1 1 0 1 0 0 1 ${\displaystyle f_{5}}$ ${\displaystyle 0101}$ ${\displaystyle {\texttt {(}}v{\texttt {)}}}$ 1 0 1 0 1 0 1 0 ${\displaystyle f_{10}}$ ${\displaystyle 1010}$ ${\displaystyle v}$ 0 1 0 1 0 1 0 1 ${\displaystyle f_{7}}$ ${\displaystyle 0111}$ ${\displaystyle {\texttt {(}}u~v{\texttt {)}}}$ 1 0 0 0 1 1 1 0 ${\displaystyle f_{11}}$ ${\displaystyle 1011}$ ${\displaystyle {\texttt {(}}u~{\texttt {(}}v{\texttt {))}}}$ 0 1 0 0 1 1 0 1 ${\displaystyle f_{13}}$ ${\displaystyle 1101}$ ${\displaystyle {\texttt {((}}u{\texttt {)}}~v{\texttt {)}}}$ 0 0 1 0 1 0 1 1 ${\displaystyle f_{14}}$ ${\displaystyle 1110}$ ${\displaystyle {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}}$ 0 0 0 1 0 1 1 1 ${\displaystyle f_{15}}$ ${\displaystyle 1111}$ ${\displaystyle {\texttt {((}}~{\texttt {))}}}$ 0 0 0 0 1 1 1 1

 ${\displaystyle {\begin{matrix}u\!:\\v\!:\end{matrix}}}$ ${\displaystyle {\begin{matrix}1100\\1010\end{matrix}}}$ ${\displaystyle f}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{11}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{10}{\texttt {)}}\\{\text{No}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{01}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}{\texttt {(}}\ell _{00}{\texttt {)}}\\{\text{No}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{00}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{01}\\{\text{Some}}~{\texttt {(}}u{\texttt {)}}\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{10}\\{\text{Some}}~u\\{\text{is}}~{\texttt {(}}v{\texttt {)}}\end{smallmatrix}}}$ ${\displaystyle {\begin{smallmatrix}\ell _{11}\\{\text{Some}}~u\\{\text{is}}~v\end{smallmatrix}}}$ ${\displaystyle f_{0}}$ ${\displaystyle 0000}$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ 1 1 1 1 0 0 0 0 ${\displaystyle f_{1}}$ ${\displaystyle 0001}$ ${\displaystyle {\texttt {(}}u{\texttt {)(}}v{\texttt {)}}}$ 1 1 1 0 1 0 0 0 ${\displaystyle f_{2}}$ ${\displaystyle 0010}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}~v}$ 1 1 0 1 0 1 0 0 ${\displaystyle f_{4}}$ ${\displaystyle 0100}$ ${\displaystyle u~{\texttt {(}}v{\texttt {)}}}$ 1 0 1 1 0 0 1 0 ${\displaystyle f_{8}}$ ${\displaystyle 1000}$ ${\displaystyle u~v}$ 0 1 1 1 0 0 0 1 ${\displaystyle f_{12}}$ ${\displaystyle 1100}$ ${\displaystyle u}$ 0 0 1 1 0 0 1 1 ${\displaystyle f_{10}}$ ${\displaystyle 1010}$ ${\displaystyle v}$ 0 1 0 1 0 1 0 1 ${\displaystyle f_{9}}$ ${\displaystyle 1001}$ ${\displaystyle {\texttt {((}}u{\texttt {,}}v{\texttt {))}}}$ 0 1 1 0 1 0 0 1 ${\displaystyle f_{6}}$ ${\displaystyle 0110}$ ${\displaystyle {\texttt {(}}u{\texttt {,}}v{\texttt {)}}}$ 1 0 0 1 0 1 1 0 ${\displaystyle f_{5}}$ ${\displaystyle 0101}$ ${\displaystyle {\texttt {(}}v{\texttt {)}}}$ 1 0 1 0 1 0 1 0 ${\displaystyle f_{3}}$ ${\displaystyle 0011}$ ${\displaystyle {\texttt {(}}u{\texttt {)}}}$ 1 1 0 0 1 1 0 0 ${\displaystyle f_{7}}$ ${\displaystyle 0111}$ ${\displaystyle {\texttt {(}}u~v{\texttt {)}}}$ 1 0 0 0 1 1 1 0 ${\displaystyle f_{11}}$ ${\displaystyle 1011}$ ${\displaystyle {\texttt {(}}u~{\texttt {(}}v{\texttt {))}}}$ 0 1 0 0 1 1 0 1 ${\displaystyle f_{13}}$ ${\displaystyle 1101}$ ${\displaystyle {\texttt {((}}u{\texttt {)}}~v{\texttt {)}}}$ 0 0 1 0 1 0 1 1 ${\displaystyle f_{14}}$ ${\displaystyle 1110}$ ${\displaystyle {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}}$ 0 0 0 1 0 1 1 1 ${\displaystyle f_{15}}$ ${\displaystyle 1111}$ ${\displaystyle {\texttt {((}}~{\texttt {))}}}$ 0 0 0 0 1 1 1 1

Table 21 provides a thumbnail sketch of the relationships discussed in this section.

 ${\displaystyle {\text{Mnemonic}}}$ ${\displaystyle {\text{Category}}}$ ${\displaystyle {\text{Classical Form}}}$ ${\displaystyle {\text{Alternate Form}}}$ ${\displaystyle {\text{Symmetric Form}}}$ ${\displaystyle {\text{Operator}}}$ ${\displaystyle {\begin{matrix}{\text{E}}\\{\text{Exclusive}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{Universal}}\\{\text{Negative}}\end{matrix}}}$ ${\displaystyle {\text{All}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle {\text{No}}~u~{\text{is}}~v}$ ${\displaystyle {\texttt {(}}\ell _{11}{\texttt {)}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{Absolute}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{Universal}}\\{\text{Affirmative}}\end{matrix}}}$ ${\displaystyle {\text{All}}~u~{\text{is}}~v}$ ${\displaystyle {\text{No}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle {\texttt {(}}\ell _{10}{\texttt {)}}}$ ${\displaystyle {\text{All}}~v~{\text{is}}~u}$ ${\displaystyle {\text{No}}~v~{\text{is}}~{\texttt {(}}u{\texttt {)}}}$ ${\displaystyle {\text{No}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~v}$ ${\displaystyle {\texttt {(}}\ell _{01}{\texttt {)}}}$ ${\displaystyle {\text{All}}~{\texttt {(}}v{\texttt {)}}~{\text{is}}~u}$ ${\displaystyle {\text{No}}~{\texttt {(}}v{\texttt {)}}~{\text{is}}~{\texttt {(}}u{\texttt {)}}}$ ${\displaystyle {\text{No}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle {\texttt {(}}\ell _{00}{\texttt {)}}}$ ${\displaystyle {\text{Some}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle {\text{Some}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle \ell _{00}}$ ${\displaystyle {\text{Some}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~v}$ ${\displaystyle {\text{Some}}~{\texttt {(}}u{\texttt {)}}~{\text{is}}~v}$ ${\displaystyle \ell _{01}}$ ${\displaystyle {\begin{matrix}{\text{O}}\\{\text{Obtrusive}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{Particular}}\\{\text{Negative}}\end{matrix}}}$ ${\displaystyle {\text{Some}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle {\text{Some}}~u~{\text{is}}~{\texttt {(}}v{\texttt {)}}}$ ${\displaystyle \ell _{10}}$ ${\displaystyle {\begin{matrix}{\text{I}}\\{\text{Indefinite}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{Particular}}\\{\text{Affirmative}}\end{matrix}}}$ ${\displaystyle {\text{Some}}~u~{\text{is}}~v}$ ${\displaystyle {\text{Some}}~u~{\text{is}}~v}$ ${\displaystyle \ell _{11}}$

## Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we'll need a number of specialized tools.  To begin, we define a higher order operator ${\displaystyle \Upsilon ,}$ called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result.  Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types.  Expressing ${\displaystyle \Upsilon }$ in that form gives the following formula.

${\displaystyle \Upsilon :\bigcup _{\ell =1,2,3}((\mathbb {B} ^{k}\to \mathbb {B} )^{\ell }\to \mathbb {B} ).}$

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”.  In the case of ${\displaystyle \Upsilon ,}$ the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms.

${\displaystyle {\begin{matrix}\Upsilon _{p}^{r}q=\Upsilon (p,q,r)\\[10pt]\Upsilon _{p}^{r}:(\mathbb {B} ^{k}\to \mathbb {B} )\to \mathbb {B} \end{matrix}}}$

The operation ${\displaystyle \Upsilon _{p}^{r}q}$ evaluates the proposition ${\displaystyle q}$ on each model of the proposition ${\displaystyle p}$ and combines the results according to the method indicated by the connective parameter ${\displaystyle r.}$  In principle, the index ${\displaystyle r}$ may specify any logical connective on as many as ${\displaystyle 2^{k}}$ arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums.  By convention, each of the accessory indices ${\displaystyle p,r}$ is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition ${\displaystyle 1:\mathbb {B} ^{k}\to \mathbb {B} }$ for the lower index ${\displaystyle p}$ and the continued conjunction or continued product operation ${\displaystyle \textstyle \prod }$ for the upper index ${\displaystyle r.}$  Taking the upper default value gives license to the following readings.

${\displaystyle {\begin{matrix}\Upsilon _{p}(q)=\Upsilon (p,q)=\Upsilon (p,q,\textstyle \prod ).\\[10pt]\Upsilon _{p}=\Upsilon (p,{\underline {~~}},\textstyle \prod ):(\mathbb {B} ^{k}\to \mathbb {B} )\to \mathbb {B} .\end{matrix}}}$

This means ${\displaystyle \Upsilon _{p}(q)=1}$ if and only if ${\displaystyle q}$ holds for all models of ${\displaystyle p.}$  In propositional terms, this is tantamount to the assertion that ${\displaystyle p\Rightarrow q,}$ or that ${\displaystyle {\texttt {(}}p{\texttt {(}}q{\texttt {))}}=1.}$

Throwing in the lower default value permits the following abbreviations.

${\displaystyle {\begin{matrix}\Upsilon q=\Upsilon (q)=\Upsilon _{1}(q)=\Upsilon (1,q,\textstyle \prod ).\\[10pt]\Upsilon =\Upsilon (1,{\underline {~~}},\textstyle \prod )):(\mathbb {B} ^{k}\ \to \mathbb {B} )\to \mathbb {B} .\end{matrix}}}$

This means ${\displaystyle \Upsilon q=1}$ if and only if ${\displaystyle q}$ holds for the whole universe of discourse in question, that is, if and only ${\displaystyle q}$ is the constantly true proposition ${\displaystyle 1:\mathbb {B} ^{k}\to \mathbb {B} .}$  The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

## References

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Quine, W.V. (1969/1981), “On the Limits of Decision”, Akten des XIV. Internationalen Kongresses für Philosophie, vol. 3 (1969). Reprinted, pp. 156–163 in Quine (ed., 1981), Theories and Things, Harvard University Press, Cambridge, MA.

## Document History

### 1995 • Oakland University • Inquiry and Analogy

 Author: Jon Awbrey November 1, 1995 Course: Engineering 690, Graduate Project Winter Term, January 1995 Supervisors: M.A. Zohdy and F. Mili Oakland University
| Version:  Draft 3.25
| Created:  01 Jan 1995
| Relayed:  01 Nov 1995
| Revised:  24 Dec 2001
| Revised:  12 Mar 2004